Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art numerical methods most rely on differentiability or continuity, such as Newton-type method, LMI method, or homotopy method. In this paper, we will build a novel theoretical framework and reveal the intrinsic algebraic structure appearing in this kind of algebraic Riccati equations. This structure guarantees that to solve them is almost as easy as to solve deterministic/classical ones, which will shed light on the theoretical analysis and numerical algorithm design for this topic.

翻译：随机代数Riccati方程（亦称有理代数Riccati方程）源于随机线性时不变系统的线性二次最优控制问题，其求解难度曾被认为较高。现有数值方法大多依赖可微性或连续性，如牛顿型方法、LMI方法或同伦方法。本文构建了一个全新的理论框架，揭示了此类代数Riccati方程所蕴含的内在代数结构。该结构确保求解此类方程几乎与求解确定性/经典方程同样简便，这将为该课题的理论分析和数值算法设计提供重要启示。